There exist infinitely many odd integers such that is composite for every . Numbers with this property are called Riesel numbers, while analogous numbers with the minus sign replaced by a plus are called Sierpiński numbers of the second kind.

The smallest known Riesel number is , but there remain 95 smaller candidates (the smallest of which is 2293) which generate only composite numbers for all which have been checked (Ribenboim 1996, p. 358; Ballinger and Keller; Riesel Sieve Project). The problem of proving or disproving that is the smallest Riesel number is sometimes known as the Riesel problem or Riesel conjecture.

Let be smallest for which is prime, then the first few values are 2, 0, 2, 1, 1, 2, 3, 1, 2, 1, 1, 4, 3, 1, 4, 1, 2, 2, 1, 3, 2, 7, ... (Sloane's A046069), and second smallest are 3, 1, 4, 5, 3, 26, 7, 2, 4, 3, 2, 6, 9, 2, 16, 5, 3, 6, 2553, ... (Sloane's A046070).

Ballinger, R. "The Riesel Problem: Definition and Status." http://www.prothsearch.net/rieselprob.html.

Ballinger, R. and Keller, W. "The Riesel Problem: Search for Remaining Candidates." http://www.prothsearch.net/rieselsearch.html.

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 357, 1996.

Riesel, H. "Några stora primtal." Elementa 39, 258-260, 1956.

Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Basel: Birkhäuser, pp. 394-398, 1994.

Riesel Sieve Project. "The Riesel Sieve Project: A Distributed Effort to Prove the Riesel Conjecture." http://www.rieselsieve.com/.

Sloane, N. J. A. Sequences A046067, A046068, A046069, and A046070 in "The On-Line Encyclopedia of Integer Sequences."

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Weisstein, Eric W. "Riesel Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RieselNumber.html